We consider the problem of dynamically maintaining (approximate) all-pairs effective resistances in separable graphs, which are those that admit an n^{c}-separator theorem for some c<1. We give a fully dynamic algorithm that maintains (1+epsilon)-approximations of the all-pairs effective resistances of an n-vertex graph G undergoing edge insertions and deletions with O~(sqrt{n}/epsilon^2) worst-case update time and O~(sqrt{n}/epsilon^2) worst-case query time, if G is guaranteed to be sqrt{n}-separable (i.e., it is taken from a class satisfying a sqrt{n}-separator theorem) and its separator can be computed in O~(n) time. Our algorithm is built upon a dynamic algorithm for maintaining approximate Schur complement that approximately preserves pairwise effective resistances among a set of terminals for separable graphs, which might be of independent interest. We complement our result by proving that for any two fixed vertices s and t, no incremental or decremental algorithm can maintain the s-t effective resistance for sqrt{n}-separable graphs with worst-case update time O(n^{1/2-delta}) and query time O(n^{1-delta}) for any delta>0, unless the Online Matrix Vector Multiplication (OMv) conjecture is false. We further show that for general graphs, no incremental or decremental algorithm can maintain the s-t effective resistance problem with worst-case update time O(n^{1-delta}) and query-time O(n^{2-delta}) for any delta >0, unless the OMv conjecture is false.
@InProceedings{goranci_et_al:LIPIcs.ESA.2018.40, author = {Goranci, Gramoz and Henzinger, Monika and Peng, Pan}, title = {{Dynamic Effective Resistances and Approximate Schur Complement on Separable Graphs}}, booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)}, pages = {40:1--40:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-081-1}, ISSN = {1868-8969}, year = {2018}, volume = {112}, editor = {Azar, Yossi and Bast, Hannah and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2018.40}, URN = {urn:nbn:de:0030-drops-95036}, doi = {10.4230/LIPIcs.ESA.2018.40}, annote = {Keywords: Dynamic graph algorithms, effective resistance, separable graphs, Schur complement, conditional lower bounds} }
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